= Solution
The exponential factor removes the first spatial derivative. Apply the <chain rule> explicitly:
$$
\partial_z\Delta\rho=e^{-z/(2H)}\left(\Psi_z-\frac{\Psi}{2H}\right),\qquad
\partial_z^2\Delta\rho=e^{-z/(2H)}\left(\Psi_{zz}-\frac{\Psi_z}{H}+\frac{\Psi}{4H^2}\right),
$$
while $\partial_t^2\Delta\rho=e^{-z/(2H)}\Psi_{tt}$. Substituting into the supplied density equation, the terms proportional to $\Psi_z$ cancel, leaving
$$
\boxed{\Psi_{tt}-c^2\Psi_{zz}+\frac{c^2}{4H^2}\Psi=0,
\qquad \omega_0=\frac{c}{2H}.}
$$
This is the <atmospheric acoustic cutoff transformation>. There is a sign error in the target equation in the original PDF, as well as in the transcription: the last term must be $+\omega_0^2\Psi$ for real positive $\omega_0$. If the displayed minus sign is retained literally, coefficient matching would require $\omega_0^2=-c^2/(4H^2)$, so it would not define a real characteristic oscillation period.
For the corrected equation, a <plane wave> $\Psi\propto e^{i(kz-\omega t)}$ has <dispersion relation>
$$
\omega^2=c^2k^2+\omega_0^2.
$$
Thus $\omega_0$ is the <acoustic cutoff frequency>: vertically propagating <acoustic waves> have $|\omega|\ge\omega_0$. The <scale height> and <sound speed> have units metres and metres per second, respectively, so $\omega_0$ has units $\mathrm{s^{-1}}$.
For an order-of-magnitude estimate in the lower <terrestrial atmosphere>, take $H\simeq8.0\ \mathrm{km}$ and $c\simeq340\ \mathrm{m\,s^{-1}}$. These are representative atmospheric values, not additional data specified by the paper. They give
$$
\omega_0\simeq2.1\times10^{-2}\ \mathrm{s^{-1}},\qquad
\boxed{\frac{2\pi}{\omega_0}=\frac{4\pi H}{c}\simeq3.0\times10^2\ \mathrm s\simeq5\ \mathrm{min}.}
$$
The <isothermal atmosphere> approximation is local: a real atmospheric <temperature> profile and the thermodynamic response of the disturbance modify the precise cutoff and period.
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